enrich
   nat
with
   sorts Iset[A];
   constants 
       ∅ : Iset[A];
   
   functions 
       . ++ . : Iset[A] × A → Iset[A] prio 9 left;
       χ : Iset[A] → A → Bool;
       { . } : (A → Bool) → Iset[A];
   
   predicates 
       . ∈ . : A × Iset[A];
       . ⊆ . : Iset[A] × Iset[A];
   
   variables 
       s : Iset[A];
       a : A;
       b : A;
       f : A → Bool;
   
   induction 
      Iset[A] generated by { . } : ((A → Bool) → Iset[A]);
   
   axioms 
      characteristic-function: χ(s)(a) ↔ a ∈ s;
      used for: s, ls;
      
      construct-set: a ∈ {f}  ↔ f(a);
      used for: s, ls;
      
      Extension: s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2);
      used for: ls;
      
      In-empty: ¬ a ∈ ∅;
      used for: s, ls;
      
      In-insert: a ∈ s ++ b ↔ a = b ∨ a ∈ s;
      used for: s, ls;
      
      Subset: s1 ⊆ s2 ↔ (∀ a. a ∈ s1 → a ∈ s2);
      used for: ls;
   
   
end enrich